I taught using a three-act math task in Cambridge last winter. The good folks at NRich posted the video so I’m highlighting some of the pedagogy behind this kind of mathematical modeling. Ask questions and share suggestions.

Act Three & Sequel

[18:36] “This guy wants to make a pyramid out of a billion pennies. And I’m curious how big that would be. Help me with that if you’re completely finished here. Or tackle some of the other questions we had up there earlier.”

[20:20] “Is that number in between your high and low from earlier? Does it fit in the range of possible numbers for you? If it didn’t we should go back and ask ourselves ‘do we trust the mathematics here?'”

But there’s another reason why students ought to see the answer to modeling tasks. (I’m not picky about answers to other tasks.) The Common Core’s modeling framework asks students to “validate the conclusions” of their models. Showing the answer acknowledges the messiness inherent to mathematical modeling and allows students to discuss possible sources of error and then account for them with newer, better models.

Make good on the promises from act one. Earlier I asked students for numbers they knew were too high and too low so I asked them here to check their answer against those numbers. I said I was curious who had the closest guess so I had to find out who did and show them some appreciation. I said I hoped we would get to everybody’s questions by the end of the day so I returned to those questions. If I fail to make good on any of those promises, I know they’ll seem awfully insincere the next time I try to make them.

Good sequels are hard to come by. The goals of the sequel task are to a) challenge students who finished quickly so b) I can help students who need my help. It can’t feel like punishment for good work. It can’t seem like drudgery. It has to entice and activate the imagination.

I have one strategy I’ll try on instinct: I flip the known and the unknown of the problem and see if the resulting question is at all interesting. In this case, I originally gave students the dimensions of the pyramid and asked for the number of pennies. So now I’ll give them the number of pennies (one billion) and ask for the dimensions. Then I try to activate their imagination around the sequel, asking “Would you be able to build it in this room? Would it punch through the ceiling?” Etc.

In some cases, the initial task just serves to set an imaginative hook for the sequel, which is much more demanding and interesting. Once students have a strong mental image of the pyramid of pennies, I can ask them to manipulate it in some flexible and interesting ways. (Nathan Kraft has written about this recently.)

What’s Missing

Formalize the math. Because I’m working with adults, I gave the math a brief treatment here. In general, act three is where the math is formalized and consolidated. Conflicting ideas are brought together and reconciled. Formal mathematical vocabulary is introduced.

Title the lesson. Lately, taking inspiration from this Japanese classroom, I ask students to provide a title that will summarize the entire lesson. Then I offer my own.

All of this happens at the end of the lesson, not the start. I’m not defining vocabulary at the start of the lesson and I’m not greeting students at the start of class with an objective on the board. Those moves make it harder for students to access the lesson, lofting interesting mathematics high up on the ladder of abstraction.

Homework

Here’s my best guess how this kind of task would look in a print-based textbook. How does it differ from the task I did in Cambridge? Try to resist easy qualifiers like, “It’s more boring,” etc. How is it more boring? How is the math different? What are the downsides? What are the upsides? (I can think of at least one.)

Your Analysis

What did you see in that clip that I didn’t talk about here? What was missing? What would you add? What would you have done differently?

As soon as I know I have all the data, the exploring side of my brain just checks out. I go straight to my brains list of formulas and start looking for ones that will fit together to solve the problem. When I don’t have the numbers yet, I can almost feel synapses firing all over my brain.

That first sentence is sure a doozy. “A pyramid is made out of layers of stacks of pennies.” If you have a picture of what that means, then sure, it makes sense, but if you don’t, it doesn’t exactly give you a lot of clarification about what it means.

I think your textbook version of the problem is missing something like “the formula for finding the sum of a sequence of square numbers is S = n(n+1)(2n+1)/6.” For whatever reason, the formula for finding the sum of a sequence of square numbers is typically not done, at least in any of the units on sequences I’ve been told to do (I always include some discussion of it anyway), and so a textbook author might be compelled to include it, forgetting all about the Internet… and/or spreadsheets.

Two quick thoughts for now (have to go help with an AP French exam . . .)

1. Your answer to “why do it” was both interesting and disappointing to me. It was interesting because I was interested in why the person originally did it. It was disappointing because my interpretation of “why do it” was from the perspective of a student asking about the task – “why do I want to calculate this number?” That didn’t get addressed. Would your answer be just because it’s inherently interesting, or because humans are curious, or what, exactly?

2. I hate to always go back to this, but where does this fit in a curriculum? Would you use this when learning about sequences? Summation? Or do you think that where you place it is irrelevant, you would just use it because it’s good? (Just struggling as a teacher of one Algebra one section trying to figure out how to incorporate the amazing stuff you do in a very limited amount of time with kids, and with a not-so-limited amount of curriculum to “cover.”)

Oh, and the obvious question that perhaps you’ve addressed elsewhere (I’m behind on my reading) – when are you going to put together the digital “textbook” – common core aligned, of course – that includes all the “upsides” and mitigates any “downsides?”

As soon as I know I have all the data, the exploring side of my brain just checks out. I go straight to my brains list of formulas and start looking for ones that will fit together to solve the problem.

When I don’t have the numbers yet, I can almost feel synapses firing all over my brain. I’m looking at the pyramid and actually seeing it. My pattern recognition parts of my brain are drawn to the splotches of newer and older coins. I question briefly whether the sides are truly equal in length, and size things up for a moment for deciding whether I buy that or not. I look at whether the edges and corners are crisp or sloppy. I’m reminded of the feel of sorting cold coins, and the smell (taste?) of copper. I think of Scrooge McDuck and Egypt and pranks and the Statue of Liberty and cash registers and the glass coin jar in my parents’ bedroom and my own attempts to count coins.

All those things happen in just a few moments, and all those connections bring me deeper into the problem. Now I really want to study this pyramid of pennies.

That first sentence is sure a doozy. “A pyramid is made out of layers of stacks of pennies.” If you have a picture of what that means, then sure, it makes sense, but if you don’t, it doesn’t exactly give you a lot of clarification about what it means.

In the lesson, you had both an image (mental image, actual video) of them building building the pyramid, as well as the close up picture (mental image, actual image) to help people understand what they pyramid actually looked like. In this problem, there’s a very zoomed out picture, that shows you what a pyramid looks like, but doesn’t give you a good idea of what _this_ pyramid looks like.

@Dan: “What’s Missing: Formalize the math. Because I’m working with adults, I gave the math a brief treatment here…” That was my only complaint about this video (as I mentioned in my comment under Act 2.) I gather that your purpose was to show how you roll re: 3-act stuff. With regards to that, I’d say: mission accomplished. If someday you would be inclined to show how you *teach math* — i.e. the boring, nitty-gritty stuff — I would be very interested to watch, as I’m sure many others would be as well.

The thing that bugged me is when you asked who was closest, 5 million or 10,000, and used subtraction to measure it. Surely it is more appropriate to use a logarithmic scale here, or, to put it another way, ratios. Anyway, for me, 5000000 is closer (the error is a factor of ~17, not ~28).

Now, whether you want to have that discussion (what do we mean by closest?) depends on the particular class, their level of mathematical knowledge, and the available time. Given the audience in the video, I surprised you did not at least mention this.

There’s a moment at the end where one of the attendees praises you for using their questions.

You kind of playfully dismiss this saying that you could have predicted the top four questions and you go on to make the point that students hate when teachers try to get them to guess the questions we want asked.

I think, though, that you too quickly dismiss one of your strongest assets. You captured the students questions in their own words before making any adjustments. That was huge. You also kept a question that would not have been on my top four and wasn’t in the top four that you mentioned: “Why would anyone do this”. The answer to this widely held question was very interesting and it rewarded students who thought they were getting a jab in.

You really did value the questions you got and you really did try to answer them. No book or software can do this.

It was disappointing because my interpretation of “why do it” was from the perspective of a student asking about the task – “why do I want to calculate this number?” That didn’t get addressed. Would your answer be just because it’s inherently interesting, or because humans are curious, or what, exactly?

One outcome of show and tell and miscellaneous questions on openers is that my classes know curiosity is king. And not just mathematical curiosity. (“What is the only country that starts with an ‘O’?” isn’t in the CCSS-M.) It’s on the class coat of arms: “get curious or get out.” That goes a long way, but then I start the task at such a low bar – questions, guesses, wrong answers, etc. – and the students have a hook buried in their cheek without really knowing how it got there. But the effect of the task definitely depends on the culture surrounding it and vice versa.

@James, I gave the math a brief treatment but it wasn’t invisible. I’d linger a bit longer on the structure of the summation notation and students would practice some decontextualized series and sequence problems later but that’s exactly how I’d treat it with students otherwise.

You use the term “boring, nitty-gritty stuff” though and I’m not sure we all have a shared understanding there. What are you referring to?

Daniel:

You captured the students questions in their own words before making any adjustments. That was huge. You also kept a question that would not have been on my top four and wasn’t in the top four that you mentioned: “Why would anyone do this”. The answer to this widely held question was very interesting and it rewarded students who thought they were getting a jab in.

You really did value the questions you got and you really did try to answer them.

Thanks. I’ll take it. Over time, it becomes easier and easier to not feel threatened by those questions, to write them down just like any other, and to return to them later.

Big ups to different Dave and Steve and their strong analysis of the textbook task. I couldn’t agree more and pushed them up to the main post.

1. You did some nice, quick work on notation. “Would anyone know what it looked like if we were multiplying” was a great question. Also liked “This tells me what I want, this tells me how to get it.” Wonder if you initially thought that there was more background knowledge on that formula. Great adjustment, if so.

2. “But don’t do nothing.” Curious: how would/did this play out in one of your old classrooms? Say a student finished at the 15:00 mark, but 80% needed until the 33:00 mark. I agree that “the billion pennies” prompt is so compelling and is probably better than the alternative (“do eight more textbook problems”).

But I had a tendency to conflate “does work quickly and correctly” with “is interested in more cognitively demanding work.”

3. “Say more about that, please.” This is a highly underrated teacher move, no? You got people talking with this prompt.

4. Love everyone’s comments (both in the video and on this thread) comparing this problem to the textbook version.

My ah-ha moment: when there’s a discussion on whether the next stack has 38 or 39 and you show them that footage of half a penny covered up. That would never, under any circumstances, be discussed or talked about in the context of that textbook problem. It would be an unchecked assumption.

The textbook problem fails as a modeling task. It’s not all bad though and may be useful for other purposes. It’s quicker. It has “high literacy demand” which, in the age of close reading, might be beneficial. Kids would have to attend to precision in the reading of words like “stack” and “layer.” There’s potential to reverse processes: A teacher could easily slide a part “b” in there that stated “What would this look like with a billion pennies?”

I agree with Karl… “when are you going to put together the digital “textbook” – common core aligned, of course – that includes all the “upsides” and mitigates any “downsides?”

How long until textbooks evolve into web-based / online / digital-only texts? A student who is actively engaged in a lesson like the one you presented in the video is so much better off than one who encounters the textbook homework problem. My best guess is 10 years.

@Dan: “You use the term “boring, nitty-gritty stuff” though and I’m not sure we all have a shared understanding there. What are you referring to?”

I’m going to answer your question with reference to the unit I just completed, leaving the pyramid question to the side for a moment. I’ve just completed a unit on Rational Functions with my Algebra 2 students. Here is a list of some of the things students learned:

1. how to add and subtract rational expressions
2. how to simplify, multiply, and divide rational expressions
3. how to simplify complex fractions
4. how to solve equations with rational expresssions
5. how to analyze the graph of a (simple) rational function

My lessons were completely devoid of pyramids, pennies, photos, videos, context, pseudo-context, or anything remotely interesting. Just plain old, boring, “nitty gritty algebra stuff.” I am not proud of this, but it’s true. I would be interested to know what a Dan Meyer version of this unit would look like. (Part of the problem here is the limitations of my own knowledge of rational functions. I can do all of the above tasks in a pure math setting, but have little knowledge of why rational functions are useful for solving interesting problems.)

Back to my main point about your lesson on pyramids: for me, the mathematical high point of the task is *developing* the formula for the sum of squares. In theory, all that “hook” you invested — showing the video, getting them to generate questions and guesses, etc — is all working towards getting them invested in doing the deep mathematics. So as I said, I was disappointed that this didn’t receive any attention. Yes, you discussed the key features of sigma notation, but that misses the point I’m driving at. Analogy: you’re teaching a lesson on the area of a triangle. The teacher can stop to make sure the students understand the *features* of the formula — what does b stand for? what does h stand for? can you point these out in the figure? etc — but if the teacher never makes the students *understand* the origins of the formula, then that is a travesty.

Maybe this is a matter of taste. I tend to favor deriving *everything,* but I’ve learned over the years that this is not necessarily best.

Summary of math required of the participants in the video:

1. Lots of modeling stuff. All important. Asking questions. Figuring out what info is needed. Making estimates. etc.

2. Plug n = 40 into formula that came “from the clear blue sky,” provided by the teacher.

So, again, if the purpose of the lesson is to practice math modeling, then great! But if the lesson were given in a context of exploring and learning about sequences and series, then it leaves a bit to be desired in that area.

@James:
I don’t mean to speak for @Dan, but I don’t think the penny modeling problem was intended to assist in deriving the formula for finding the sum of squares.
However, I too would love to hear or see what a @Dan version of the algebra 2 unit described above would look like.
Finally, I agree with you that it is important that we turn at least some of our focus to deriving the formulas we use. I feel the Common Core calls for us to do so. Your example of the area of a triangle is perfect. I just completed a unit in my geometry class. I had questions on my assessments that asked students to explain how some of the area formulas were derived, almost like informal proofs – triangles, parallelograms, trapezoids, etc. I’ve never asked students to explain where these formulas come from before. I was very happy with the results.

I don’t see any part of learning something “new” in math, including the 5 items on rational functions, as boring and nitty gritty at all. What is boring is having to repeatedly do procedural stuff. There is necessary procedural practice, then there is overkill procedural practice. I think kids would WANT to learn all that rational expression/equation stuff if there was a HOOK for them to learn it. “Work” problems might provide the hook to learn how to solve rational equations, although I don’t teach it using equations, I do it visually first (http://fawnnguyen.com/2012/12/11/20121211.aspx)

We were talking about irrational numbers recently in my algebra 1 class, and I started by asking them about the diagonal of a 1×1 square. The fact that its diagonal is sqr(2) is mind blowing — this shook the followers of Pythagoras, they HID this fact because they were scared of it (just as the Greeks banned the idea of zero)! Kids love to hear stories. I’d done a lesson on constructing irrational numbers on the real number line (without telling them how), but they remembered this and it made all the subsequent “boring” stuff not so boring because there was a reason for learning it.

I also picked up on this. As you point out, the arithmetic mean is nowhere near, but in fact the geometric mean is much closer:

sqrt(10,000*5,000,000)=223,606, not so far off 287,820.

I guess this is because when estimating, people calibrate their estimate by multiplying (e.g. by up to 10) rather than adding a few centimetres as might be the case if you were estimating someone’s height?

First of all a HUGE thank you to Dan for his session at the NRich/PRIMAS day in Cambridge in March. It was hugely inspirational & I finally “had a go” myself today using the Yellow Starbursts resources.

I have a tricky Year 9 (13-14 yrs) class and was rather apprehensive as they don’t work well with open ended tasks where the structure is not immediately apparent. They were great! The maths talk & debate that came out of the lesson was brilliant. They lost their way when it came to the actual calculations but nevertheless their involvement in trying to solve the problem was fantastic. I’m looking forward to the next time I use this type of lesson.

[…] along with the amazing work that Dave Major has done, suggests a three act structure that builds on Dan Meyer’s original three act sequence. It starts with the same basic premise of Act 1 – a simple, engaging, and non-threatening […]